I'm using the following script:

<?php
 mysql_connect("inspectlettest.ce0ohcvwmist.us-east-1.rds.amazonaws.com", "username", "password") or die(mysql_error());
 echo "Connected to MySQL<br />";
?>

The net server is oddly coming back a 500 Internal Server Error.

I'm able to connect with the mysql server via terminal in the instance all right.

Any ideas?

When you are getting an interior server error when performing a php script, the first thing ought to be to have more details about this error. On Ubuntu, you should check following apache log file:

/var/log/apache2/error.log

It's possible that it'll say something of that nature:

[...] PHP Fatal error:  Call to undefined function mysql_connect() in [...]

If you notice it, it's possible that some needed packages aren't placed on the body. Basically remember properly, you'll need a minimum of mysql-client and php-mysql packages. Restarting the apache server can also be necessary:

$ sudo apt-get install mysql-client
$ sudo apt-get install php-mysql
$ sudo /etc/init.d/apache2 restart

In case your error message differs, please combine it with the question, it'll make assisting you simpler.